See the attached file for the correct version since it doesn’t show right on the website
1. Consider the following simple linear regression modelYi =?0+?1Xi+?i
with ?i independent normall-distributed errors with E [?i] = 0 and Var[?i] =?2, for i = 1,…,n (e.g. ?i ? N(0,?2)). A common use of the regression model is to estimate the value of the outcome Y conditional upon a given valueofX. AnaturalestimateofE[Y|X =Xh]isgivenbyYˆh:
Yˆh = ?ˆ0+?ˆ1Xh
= Y? ? ?ˆ 1 X? + ?ˆ 1 X h= Y?+?ˆ1(Xh?X?)
(a) FindtheexpectationofYˆh.IsYˆhanunbiasedestimatorofE[Y|X=Xh]? (b) Show that the variance of Yˆh is given by
n ? ni = 1 ( X i ? X? ) 2(Hint:youwillneedtousethefactthatCovY?,?ˆ =0.
(c) WhatisthedistributionofYˆhundertheassumptionsofthemodelgiven above?
(d) Suppose we estimate ?2 by s2 = RSS , derive the distribution ofn?2
1 (X?X?)2 ?2 + h
Yˆ h ? E [ Y | X = X h ]1 (Xh?X?)2
s2 n + n ? 2?i=1 (Xi ?X )
(Hint: you can use the fact that RSS ? ?2 that we mentioned in class